where \(h:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\)  and \(j:\Gamma _2\times \mathbb {R}\rightarrow \mathbb {R}\)  are locally Lipschitz w.r.t. the second variable and \(\partial _C^2 h(x,t)\)  stands for the Clarke subdifferential (see, e.g., Clarke [1]}) of the mapping \(t\mapsto g(x,t)\) . The fact that we opted to partition \(\Gamma \)  into three parts for problem \((P)\) , although it poses no mathematical difficulties, comes from the fact that \((P)\)  can serve as a model for the antiplane shear deformation of cylinders, of Hencky-type material, in contact with a rigid foundation as it can be seen in the following subsection.
Due to the fact that convex functions are in fact locally Lipschitz on the interior of their effective domain and the convex and Clarke subdifferentials coincide in this case, problem \((P_1)\)  is in fact a bit more general than \((P)\) . The reasoning for choosing the convex sudifferential here is the following: in [2]} the authors show (through a different approach) that \((P_1)\)  possesses at least one weak solution under the key assumption that the Clarke subdifferential of \(h\)  satisfies a growth condition of the following type
\(|\partial _C^2 h(x,t)|\le a_1(x)+c_1\psi (|t|), \ \mbox{ for a.e. }x\in \Omega \mbox{ and all }t\in \mathbb {R},\) 
